# Quantum phase transitions in a charge-coupled Bose-Fermi Anderson model

###### Abstract

We study the competition between Kondo physics and dissipation within an Anderson model of a magnetic impurity level that hybridizes with a metallic host and is also coupled, via the impurity charge, to the displacement of a bosonic bath having a spectral density proportional to . As the impurity-bath coupling increases from zero, the effective Coulomb interaction between two electrons in the impurity level is progressively renormalized from its repulsive bare value until it eventually becomes attractive. For weak hybridization, this renormalization in turn produces a crossover from a conventional, spin-sector Kondo effect to a charge Kondo effect. At particle-hole symmetry, and for sub-Ohmic bath exponents , further increase in the impurity-bath coupling results in a continuous, zero-temperature transition to a broken-symmetry phase in which the ground-state impurity occupancy acquires an expectation value . The response of the impurity occupancy to a locally applied electric potential features the hyperscaling of critical exponents and scaling that are expected at an interacting critical point. The numerical values of the critical exponents suggest that the transition lies in the same universality class as that of the sub-Ohmic spin-boson model. For the Ohmic case , the transition is instead of Kosterlitz-Thouless type. Away from particle-hole symmetry, the quantum phase transition is replaced by a smooth crossover, but signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies.

###### pacs:

75.20.Hr, 71.10.Hf, 73.43.Nq, 05.10.Cc## I Introduction

Quantum impurity models have intrigued physicists for more than half a century.Hewson:93 In recent years, the focus has largely been on models that exhibit quantum phase transitions (QPTs). Strictly, these are boundary QPTs at which only a subset of system degrees of freedom becomes critical.Vojta:06 Boundary QPTs not only serve as prototypes for the bulk QPTs encountered (or postulated to exist) in many strongly correlated systems,Sondhi:97 ; Sachdev:99 but in certain cases they are amenable to controlled realization in quantum-dot setups.qd-qpt-expts

Of great current interest are dissipative quantum impurity models that describe a dynamical local degree of freedom coupled to one or more bosonic modes representing a frictional environment. Experiments on single-molecule transistorsset-expts have drawn attention to transport through nanodevices featuring electron-phonon interactions as well as local electron-electron interactions. The essential physics of these experiments seems to be captured in variantsAH-qdot1 ; Cornaglia:04+05 ; AH-qdot2 ; Zitko:06 ; AH-qdot3 ; Dias:09 of the Anderson-Holstein model, which augments the Anderson impurity modelAnderson:61 with a Holstein coupling of the impurity occupancy to a local (nondispersive) phonon mode. The Anderson-Holstein model has been studied since the 1970s in connection with the phenomenon of mixed valence,mixed-valence ; Hewson:02+Jeon:03 ; Lee:04 and has also been adapted to treat the effect of negative- tunneling centers on superconductivity.negative-U ; Schuttler:88 The many theoretical approaches that have been applied to these models have yielded general agreement that phonons serve to reduce the effective Coulomb repulsion between electrons in the impurity level, or even to produce an attractive net electron-electron interaction. Most challenging has been the study of simultaneous strong Coulomb repulsion and strong electron-phonon coupling. Here, the most robust solutions have been provided by an extension of the numerical renormalization-group (NRG) technique, long established as a reliable tool for tackling pure-fermionic quantum impurity problems.Wilson:75 ; Krishna-murthy:80 ; Bulla:08 NRG studiesHewson:02+Jeon:03 ; Cornaglia:04+05 ; Zitko:06 have shown that in the one-channel Anderson-Holstein model, descriptive of a single molecule coupled symmetrically to source and drain leads, increasing the phonon coupling from zero results in a smooth crossover from a conventional Kondo effect, involving conduction-band screening of the impurity spin degree of freedom, to a predominantly charge Kondo effect in which it is the impurity “isospin” or deviation from half-filling that is quenched by the conduction band. However, even for very strong electron-phonon couplings, the ground state remains a many-body Kondo singlet and there is no QPT. By contrast, a two-channel model describing a single-molecular transistor with a center-of-mass vibrational mode exhibits a line of QPTs manifesting the critical physics of the two-channel Kondo model.Dias:09

An even greater theoretical challenge is posed by quantum impurities coupled to dispersive bosons. A canonical example is the spin-boson model, Leggett:87+Weiss:99 which describes tunneling within a two-state system coupled to a bosonic bath. The model has many proposed applications, including frictional effects on biological and chemical reaction rates,Garg:85+Onuchic:87+Evans:95 cold atoms in a quasi-one-dimensional optical trap,Recati:05 a quantum dot coupled to Luttinger-liquid leads,LeHur:05 and study of entanglement between a qubit and its environment.entanglement ; LeHur:07 In many cases, the dissipative bosonic bath can be described by a spectral density [formally defined in Eq. (II.1) below] that is proportional to at low frequencies . The spin-boson model with an Ohmic () bath has long been knownLeggett:87+Weiss:99 to exhibit a Kosterlitz-Thouless QPT between delocalized and localized phases. The existence of a QPT for sub-Ohmic () baths was for some years the subject of debate.Leggett:87+Weiss:99 ; Spohn:85+Kehrein:96 However, clear evidence for a continuous QPT has been provided by the NRG,Bulla:03+05 ; Anders:07 ; LeHur:07 by perturbative expansion in about the delocalized fixed point,Vojta:05 and through exact-diagonalization calculations.Alvermann:09

Theoretical activity has also centered on the Bose-Fermi Kondo (BFK) model,Sengupta:00 in which an impurity spin- degree of freedom is coupled both to a fermionic band of conduction electrons and to one or more bosonic baths. BFK models arise in the context of unconventional heavy-fermion quantum criticality treated within extended dynamical mean-field theory (extended DMFT) (Ref. lcqpt, ) and have also been proposed to describe quantum dots coupled either to a noisy environmentLeHur:04+Li:04+Li:05+Borda:05 or to ferromagnetic leads.Kirchner:05+08 Studies of BFK models having different spin rotation symmetry—SU(2), XY, or Ising—employing either expansionZhu:02+Zarand:02 in or the NRG (Refs. Glossop:05, and Glossop:07, ) have found continuous QPTs between phases exhibiting the Kondo effect and localized phases in which impurity spin flips are suppressed by the coupling to the bosonic bath(s). For exponents , most evidence suggests that the continuous QPTs of the spin-boson and of Ising-anisotropic BFK models are equivalent. QPTs outside the spin-boson universality class have been identified in dissipative models featuring a pseudogap in the electronic density of states. Chung:07+Glossop:08

In this paper, we combine the themes outlined in the preceding paragraphs by investigating a charge-coupled Bose-Fermi Anderson (BFA) model in which the impurity not only hybridizes with conduction-band electrons but also is coupled, via its electron occupancy, to a bath representing acoustic phonons or other bosonic degrees of freedom whose dispersion extends to zero energy. The model was introduced more than 30 years agoRiseborough:77 ; Haldane:77a ; Haldane:77b in connection with the mixed-valence problem. A spinless version of the model was also discussed in the same context.Hewson:80 More recently, very similar models have been shown to arise as effective impurity problems in the extended DMFT for one- and two-band extended Hubbard models.Smith:99 ; Smith:00 Hitherto, only limited progress has been made toward understanding the physics of such models, and we are aware of no study of their possible QPTs.

Our NRG study of the charge-coupled BFA model with bosonic baths characterized by exponents reveals a crossover with increasing electron-boson (e-b) coupling from a spin Kondo effect to a charge Kondo effect, very similar to that noted previously in the Anderson-Holstein model.Hewson:02+Jeon:03 ; Cornaglia:04+05 ; Zitko:06 However, under conditions of strict particle-hole symmetry, further increase in the e-b coupling leads to complete suppression of Kondo physics at a quantum critical point. Beyond the critical e-b coupling lies a localized phase in which charge fluctuations on the impurity site are frozen. For sub-Ohmic baths (), the QPT is continuous and the numerical values of the critical exponents describing the response of the impurity charge to a locally applied electric potential demonstrate that the transition belongs to the same universality class as that of the spin-boson and Ising BFK models. For Ohmic baths (corresponding to ), the QPT is found to be of Kosterlitz-Thouless type. Particle-hole asymmetry acts in a manner analogous to a magnetic field at a conventional ferromagnetic ordering transition, smearing the discontinuous change in the ground-state as a function of e-b coupling into a smooth crossover. Signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies.

It is important to note that questions have been raised as to whether or not the NRG method reliably captures the quantum critical behavior of the spin-boson and Ising BFK models for bath exponents . It is a standard beliefSondhi:97 ; Sachdev:99 that the low-energy behavior near a quantum phase transition in spatial dimensions is equivalent to that of a classical transition in dimensions, where is the dynamical exponent. In the case of the spin-boson and Ising BFK models, for which and , the corresponding classical system is a one-dimensional Ising chain with long-ranged interactions that decay for large separations like . The Ising chain is known to possess an interacting critical point for , but to exhibit a mean-field transitionFisher:72+Luijten:96+Luijten:97 for . By contrast, NRG studies of the spin-bosonVojta:05 and Ising BFK (Refs. Glossop:05, and Glossop:07, ) models have found non-mean-field behavior extending over the entire range , leading to a claim of breakdown of the quantum-to-classical mapping.Vojta:05 This claim has recently been contradicted by continuous-time Monte CarloWinter:09 and exact diagonalizationAlvermann:09 studies. Debate is ongoing concerning the interpretation of these various results.Winter:09 ; Kirchner:09+Vojta:09 The eventual resolution of this debate may determine the validity of the small subset of our NRG results that concerns the critical exponents of the charge-coupled BFA model with bath exponents . There is every reason to believe that the remaining results are physically sound.

The rest of this paper is organized as follows. Section II introduces the charge-coupled BFA Hamiltonian and summarizes the NRG method used to solve the model. Section III contains a preliminary analysis of the model, focusing on the bosonic renormalization of the effective electron-electron interaction within the impurity level. Numerical results for the symmetric model with sub-Ohmic () dissipation are presented and interpreted in Sec. IV. Section V treats the symmetric model with Ohmic () dissipation. Section VI discusses the effects of particle-hole asymmetry. The paper’s conclusions are presented in Sec. VII.

## Ii Model and Solution Method

### ii.1 Charge-coupled Bose-Fermi Anderson Hamiltonian and related models

In this work, we investigate the charge-coupled Bose-Fermi Anderson model described by the Hamiltonian

(1) |

where

(2) | ||||

(3) | ||||

(4) | ||||

(5) | ||||

(6) |

Here, annihilates an electron of spin component (or ) and energy in the impurity level, , , and is the Coulomb repulsion between two electrons in the impurity level. is the hybridization between the impurity and a conduction-band state of energy annihilated by fermionic operator , and characterizes the coupling of the impurity occupancy to bosons in an oscillator state of energy annihilated by operator . is the number of unit cells in the host metal and, hence, the number of inequivalent values. Correspondingly, is the number of oscillators in the bath, and the number of distinct values of . Without loss of generality, we take and to be real and non-negative. Throughout the paper, we drop all factors of the reduced Planck constant , Boltzmann’s constant , the impurity magnetic moment , and the electronic charge .

To focus on the most interesting physics of the model, we assume a constant hybridization and a flat conduction-band density of states (per unit cell, per spin- orientation)

(7) |

defining the hybridization width . The bosonic bath is completely specified by its spectral density, which we take to have the pure power-law form

(8) |

characterized by an upper cutoff , an exponent that must satisfy to ensure normalizability, and a dimensionless prefactor . In this paper, we present results only for the case in which the bath and band share a common cutoff. We also adopt the convention that is held constant while one varies , which we term the electron-boson (e-b) coupling. It should be emphasized, though, that the key features of the model are a nonvanishing Fermi-level density of states and the asymptotic behavior for . Relaxing any or all of the remaining assumptions laid out in this paragraph will not alter the essential physics of the model, although it may affect nonuniversal properties, such as the locations of phase boundaries.

For many purposes, it is convenient to rewriteKrishna-murthy:80 the impurity part of the Hamiltonian (dropping a constant term )

(9) |

where . Most of the results presented below were obtained for the symmetric model characterized by or , for which the impurity states and are degenerate in energy. Section VI addresses the behavior of the asymmetric model.

In any realization of involving coupling of acoustic phonons to a magnetic impurity or a quantum dot, the value of the bath exponent will depend on the precise interaction mechanism. However, phase space considerations suggest that any such system will lie in the super-Ohmic regime . Models closely related to have also been considered in the context of extended DMFT,Smith:00 ; edmft a technique for systematically incorporating some of the spatial correlations that are omitted from the conventional DMFT of lattice fermions.Georges:95 Extended DMFT maps the lattice problem onto a quantum impurity problem in which a central site interacts with both a fermionic band and one or more bosonic baths, the latter representing fluctuating effective fields due to interactions between different lattice sites. The charge-coupled BFA model serves as the mapped impurity problem for various extended Hubbard models with nonlocal density-density interactions.Smith:99 ; Smith:00 In these settings, the effective bath exponent is not known a priori, but is determined through self-consistency conditions that ensure that the central site is representative of the lattice as a whole. The extended DMFT treatment of other lattice modelslcqpt gives rise to exponents , and we expect this also to be the case for the extended Hubbard models.

At the Hartree-Fock level,Haldane:77a the impurity properties of Hamiltonian (1) are identical to those of the Anderson-Holstein Hamiltonian,

(10) |

which augments the well-studied Anderson impurity model,Anderson:61

(11) |

with a Holstein coupling of the impurity charge to a single phonon mode of energy . At several points in the sections that follow, we compare and contrast our results for with those obtained previously for .

### ii.2 Numerical renormalization-group method

We solve the charge-coupled BFA model using the NRG method,Wilson:75 ; Krishna-murthy:80 ; Bulla:08 as recently extended to treat models involving both dispersive bosons and dispersive fermions.Glossop:05 ; Glossop:07 The full range of conduction-band energies (bosonic-bath energies ) is divided into a set of logarithmic intervals bounded by () for , where is the Wilson discretization parameter. The continuum of states within each interval is replaced by a single state, namely, the particular linear combination of band (bath) states within the interval that enters (). The discretized model is then transformed into a tight-binding form involving two sets or orthonormalized operators: (i) (, 1, 2, ) constructed as linear combinations of all having ; and (ii) (, 1, 2, ) mixing all such that . This procedure maps the last four parts of Hamiltonian (1) to

(12) | |||

(13) | |||

(14) | |||

(15) |

Here, , while the remaining coefficients , , , and , which include all information about the conduction-band density of states and the bosonic spectral density , are calculated via Lanczos recursion relations.Glossop:07 For a particle-hole-symmetric density of states such as that in Eq. (7), for all .

The coefficients in Eq. (12) vary for large as , while and entering Eq. (13) vary for large as . Therefore, the problem can be solved iteratively by diagonalization of a sequence of Hamiltonians (, 1, 2, ) describing tight-binding chains of increasing length. At iteration , Eq. (12) is restricted to , while Eq. (13) is limited to . The spirit of the NRG is to treat fermions and bosons of the same energy scale at the same iteration. Since the bosonic coefficients decay with site index twice as fast as the fermionic coefficients, after a few iterations the iterative procedure requires extension of the bosonic chain only for every second site added to the fermionic chain. In this work, we have chosen for simplicity to work with a single high-energy cutoff scale . It is then convenient to add to the bosonic chain at every even-numbered iteration, so that the highest-numbered bosonic site is , where is the greatest integer less than or equal to .

The NRG method relies on two additional approximations. Even for pure-fermionic problems, it is not feasible to keep track of all the eigenstates because the dimension of the Fock space increases rapidly as we add sites to the chains. Therefore, only the lowest lying many-particle states can be retained after each iteration. The presence of bosons adds the further complication that the Fock space is infinite-dimensional even for a single-site chain, making it necessary to restrict the maximum number of bosons per chain site to a finite number . Provided that and are chosen to be sufficiently large (as discussed in Sec. IV.1), the NRG solution at iteration provides a good account of the impurity contribution to physical properties at temperatures and frequencies of order .

Hamiltonian (1) commutes with the total spin- operator

(16) |

the total spin-raising operator

(17) |

and the total “charge” operator

(18) |

which measures the deviation from half-filling of the total electron number. One can interpret

(19) |

as the generators of an SU(2) isospin symmetry (originally dubbed “axial charge” in Ref. Jones:88, ). Since , the charge-coupled BFA model does not exhibit full isospin symmetry. However, this symmetry turns out to be recovered in the asymptotic low-energy behavior at certain renormalization-group fixed points.

As described in Ref. Krishna-murthy:80, , the computational effort required for the NRG solution of a problem can be greatly reduced by taking advantage of these conserved quantum numbers. In particular, it is possible to obtain all physical quantities of interest while working with a reduced basis of simultaneous eigenstates of , , and with eigenvalues satisfying . With one exception noted in Sec. IV.7, any value specified below represents the number of retained multiplets, corresponding to a considerably larger number of states.

Even when advantage is taken of all conserved quantum numbers, NRG treatment of the charge-coupled BFA model remains much more demanding than that of the Anderson model [Eq. (11)] or the Anderson-Holstein model [Eq. (10)]. Being nondispersive, the bosons in the last model enter only the atomic-limit Hamiltonian , allowing solution via the standard NRG iteration procedure. For Bose-Fermi models such as , the need to extend a bosonic chain as well as a fermionic one at every even-numbered iteration , expands the basis of from states to states, and multiplies the CPU time by a factor . Since we typically use or 12 in our calculations, the increase in computational effort is considerable.

The choice of value for the NRG discretization parameter involves trade-offs between discretization error (minimized by taking to be not much greater than 1) and truncation error (reduced by working with ). Experience from other problemsIngersent:02 ; Glossop:05 ; Glossop:07 indicates that critical exponents can be determined very accurately using quite a large . Most of the results presented in the remaining sections of the paper were obtained for , with being employed in the calculation of the impurity spectral function. For convenience in displaying these results, we set and omit all factors of and .

## Iii Preliminary Analysis

We begin by examining the special cases in which the impurity level is decoupled either from the conduction band or from the bosonic bath. Understanding these cases allows us to establish some expectations for the behavior of the full model described by Eq. (1).

### iii.1 Zero hybridization

If one sets in Eq. (1), then the conduction band completely decouples from the remaining degrees of freedom and can be dropped from the model, leaving the zero-hybridization model

(20) |

The Fock space separates into sectors of fixed impurity occupancy (, 1, or 2), within each of which the Hamiltonian can be recast, using displaced-oscillator operators

(21) |

in the trivially solvable form

(22) |

where

(23) |

The bosons act on the impurity to reduce the Coulomb interaction from its bare value to an effective value

(24) |

For the bath spectral density in Eq. (II.1) with , one finds that for any nonzero e-b coupling , and the singly occupied impurity states drop out of the problem. For the remainder of this section, however, we will instead focus on bath exponents , for which Eqs. (II.1) and (24) give

(25) |

For weak e-b couplings, is positive and the ground state of lies in the sector where the impurity has a spin component . However, is driven negative for sufficiently large , placing the ground state in the sector or where the impurity is spinless but has a charge (relative to half filling) of or .

Figure 1 illustrates this renormalization of the Coulomb interaction for the symmetric model (), in which the and states always have the same energy. In this case, all four impurity states become degenerate at a crossover e-b coupling

(26) |

The impurity contributions to physical properties at this special point, which is characterized by effective parameters , are identical to those at the free-orbital fixed pointKrishna-murthy:80 of the Anderson model.

For the general case of an asymmetric impurity, the sectors and 2 have a ground-state energy difference for any value of . The overall ground state of Eq. (20) is a doublet (, ) for small e-b couplings, crossing over to a singlet ( for , or for ) for large . At , a point of three-fold ground-state degeneracy, the impurity contributions to low-temperature () physical properties are identical to those at the valence-fluctuation fixed pointKrishna-murthy:80 of the Anderson model.

Using the NRG with only a bosonic chain [Eq. (13)] coupled to the impurity site, we have confirmed the existence for of a simple level crossing from a spin-doublet ground state for to a charge-doublet ground state for . In the former regime, the bosons couple only to the high-energy (, 2) impurity states, so the low-lying spectrum is that of free bosons obtained by diagonalizing given in Eq. (13). Here, NRG truncation plays a negligible role provided that one works with (say).

For , the low-lying bosonic excitations should, in principle, correspond to noninteracting displaced oscillators having precisely the same spectrum as the original bath. However, the occupation number in the ground state of Eq. (22) obeys a Poisson distribution with mean . Thus, the total number of bosons corresponding to operators satisfying takes a mean value

(27) |

The bath states in the th interval are represented by NRG chain states , with the greatest weight being borne by state . Thus, a faithful representation of the displaced-oscillator spectrum requires inclusion of states having up to several times ; based on experience with the Anderson-Holstein model,Hewson:02+Jeon:03 one expects to suffice. Given that , it is feasible to meet this condition as so long as the bath exponent satisfies . Indeed, for Ohmic and super-Ohmic bath exponents, the NRG spectrum for not too much greater than is found to be numerically indistinguishable from that for . For , by contrast, the restriction leads, for and large iteration numbers, to an artificially truncated spectrum that cannot reliably access the low-energy physical properties. Nonetheless, observation of this “localized” bosonic spectrum serves as a useful indicator, both in the zero-hybridization limit and in the full charge-coupled BFA model, that the effective e-b coupling remains nonzero.

Another interpretation of Eq. (III.1) is that at the energy scale characteristic of interval , the e-b coupling takes an effective value governed by the renormalization-group equation

(28) |

which implies that the e-b coupling is irrelevant for , marginal for , and relevant for . While the NRG method is capable of faithfully reproducing the physics of for arbitrary renormalizations of , , and , its validity is restricted to the region

(29) |

For and , as used in most of our calculations, the upper limit on the “safe” range of varies from 1.7 for to for .

We now focus on the value of the crossover e-b coupling determined using the NRG approach. Figure 2 shows for five different bosonic bath exponents that has an almost linear dependence on the NRG discretization in the range . We believe that the rise in with reflects a reduction in the effective value of arising from the NRG discretization. It is knownKrishna-murthy:80 that in NRG calculations for fermionic problems, the conduction-band density of states at the Fermi energy takes an effective value

(30) |

where

(31) |

The general trend of the data in Fig. 2 is consistent with there being an analogous reduction of the bosonic bath spectral density that requires the replacement of by

(32) |

when extrapolating NRG results to the continuum limit . However, we have not obtained a closed-form expression for .

Table 1 lists values extrapolated from the data plotted in Fig. 2. For , these values are in good agreement with Eq. (26). For , however, the extrapolated value of lies significantly above the exact value, indicating that for given the NRG underestimates the bosonic renormalization of . This is most likely another consequence of truncating the basis on each site of the bosonic tight-binding chain.

0.2 | 0.4 | 0.6 | 0.8 | 1.0 | |
---|---|---|---|---|---|

0.177 | 0.251 | 0.307 | 0.355 | 0.396 | |

0.188(4) | 0.250(2) | 0.307(2) | 0.355(2) | 0.397(3) |

In analyzing our NRG results for the full charge-coupled BFA model, we attempt to compensate for the effects of discretization and truncation by replacing Eq. (25) by

(33) |

Here, is not the theoretical value predicted in Eq. (26), but rather is obtained from runs carried out for but otherwise using the same model and NRG parameters as the data that are being interpreted.

### iii.2 Zero electron-boson coupling

For , the bosonic bath decouples from the electronic degrees of freedom, which are then described by the pure Anderson model. In this section, we briefly review aspects of the Anderson model that will prove important in interpreting results for the charge-coupled BFA model. For further details concerning the Anderson model, see Refs. Hewson:93, and Krishna-murthy:80, .

For any , and for any and (whether positive, negative, or zero), the stable low-temperature regime of the Anderson model lies on a line of strong-coupling fixed points corresponding to . At any of these fixed points, the system is locked into the ground state of the atomic Hamiltonian , and there are no residual degrees of freedom on the impurity site or on site of the fermionic chain; the NRG excitation spectrum is that of the HamiltonianKrishna-murthy:80

(34) |

The coefficients are identical to those entering [Eq. (12)], except that here . Note that in Eq. (34), the sum over begins at 1 rather than 0.

As shown in Ref. Krishna-murthy:80, , the strong-coupling fixed points of the Anderson model are equivalent—apart from a shift of 1 in the ground-state charge defined in Eq. (18)—to the line of frozen-impurity fixed points corresponding to , , with NRG excitation spectra described by

(35) |

The mapping between alternative specifications of the same fixed-point spectrum isKrishna-murthy:80

(36) |

where [see Eq. (30)] is the effective conduction-band density of states.

The fixed-point potential scattering is related to the ground-state impurity charge via the Friedel sum rule,

(37) |

For , one finds that

(38) |

where is defined in Eq. (31).

Even though the stable fixed point of the Anderson model for any is one of the strong-coupling fixed points described above, the route by which such a fixed point is reached can vary widely, depending on the relative values of , , and . For our immediate purposes, it suffices to focus on the symmetric () model, for which there is a single strong-coupling fixed point corresponding to or . If the on-site Coulomb repulsion is strong enough that the system enters the local-moment regime (), then it is possible to perform a Schrieffer-Wolff transformationSchrieffer:66 that restricts the system to the sector and reduces the Anderson model to the Kondo model described by the Hamiltonian

(39) |

where

(40) |

The stable fixed point is approached below an exponentially small Kondo temperature when the spin-flip processes associated with the term in cause the effective values of and to renormalize to strong coupling, resulting in many-body screening of the impurity spin.

Motivated by the discussion in Sec. III.1, we also consider the case of strong on-site Coulomb attraction. In the local-charge regime (), a canonical transformation similar to the Schrieffer-Wolff transformation restricts the system to the sectors and , and maps the Anderson model onto a charge Kondo model described by the Hamiltonian

(41) |

where

(42) |

In this case, the stable fixed point is approached below an exponentially small (charge) Kondo temperature when the charge-transfer processes associated with the term in cause the effective values of and to renormalize to strong coupling, resulting in many-body screening of the impurity isospin degree of freedom [associated with the -operator terms in Eqs. (19)].

Between the opposite extremes of large positive and large negative is a mixed-valence regime in which interactions play only a minor role. Here, the stable fixed point is approached below a temperature of order when the effective value of scales to strong coupling, signaling strong mixing of the impurity levels with the single-particle states of the conduction band.

### iii.3 Expectations for the full model

Insight into the behavior of the full charge-coupled BFA model described by Eqs. (1)–(6) can be gained by performing a Lang-FirsovLang:62 transformation with

(43) |

The transformation eliminates , leaving

(44) |

where is as defined in Eqs. (23) and (24), and

(45) |

In addition to renormalizing the impurity interaction from to entering , the e-b coupling causes every hybridization event to be accompanied by the creation and annihilation of arbitrarily large numbers of bosons.

In the case of the Anderson-Holstein model [Eq. (10)], various limiting behaviors are understood.Schuttler:88 In the instantaneous limit , the bosons adjust rapidly to any change in the impurity occupancy; for , the physics is essentially that of the Anderson model with , while for , there is also a reduction from to in the rate of scattering between the and sectors, reflecting the reduced overlap between the ground states in these two sectors. In the adiabatic limit , the phonons are unable to adjust on the typical time scale of hybridization events, and neither nor undergoes significant renormalization.

Similar analysis for the charge-coupled BFA model is complicated by the presence of a continuum of bosonic mode energies , only some of which fall in the instantaneous or adiabatic limits. Nonetheless, we can use results for the cases (Sec. III.1) and (Sec. III.2), as well as those for the Anderson-Holstein model, to identify likely behaviors of the full model. Specifically, we focus here on the evolution with decreasing temperature of the effective Hamiltonian describing the essential physics of the symmetric () model at the current temperature. This effective Hamiltonian is obtained under the assumption that real excitations of energy above the ground state —where is a number around 5, say—make a negligible contribution to the observable properties, and thus can be integrated from the problem.

Based on the preceding discussion, one expects that at high temperatures , the physics of the charge-coupled BFA model will be very similar to that of the Anderson model with replaced by , where

(46) |

Note that is identical to defined in Eq. (24). For the bath spectral density in Eq. (II.1) with ,

(47) |

When analyzing NRG data, we instead use

(48) |

where is the empirically determined value discussed in connection with Eq. (33).

If, upon decreasing the temperature to some value , the system comes to satisfy , then it should enter a local-moment regime described by the effective Hamiltonian with the exchange couplings in [Eq. (39)] determined by Eq. (40) with , similar to what is found in the Anderson-Holstein model.Cornaglia:04+05 Since they couple only to the high-energy sectors and that are projected out during the Schrieffer-Wolff transformation, the bosons should play little further role in determining the low-energy impurity physics. The outcome should be a conventional Kondo effect where the e-b coupling contributes only to a renormalization of the Kondo scale .

If, instead, at some the system satisfies , then it should enter a local-charge regime described by the effective Hamiltonian

(49) |

Based on the behavior of the Anderson-Holstein model,Cornaglia:04+05 one expects in [Eq. (41)] to be determined by Eq. (42) with , but with exponentially depressed due to the aforementioned reduction in the overlap between the ground states of the and sectors. The bosons couple to the low-energy sector of the impurity Fock space, so they have the potential to significantly affect the renormalization of and upon further reduction in the temperature. In particular, the term in , which favors localization of the impurity in a state of well-defined or , directly competes with the double-charge transfer term that is responsible for the charge Kondo effect of the negative- Anderson model. This nontrivial competition gives rise to the possibility of a QPT between qualitatively distinct ground states of the charge-coupled BFA model.

Between these extremes, the system can enter a mixed-valence regime of small effective on-site interaction. In this regime, one must retain all the impurity degrees of freedom of the charged-coupled BFA model. The impurity-band hybridization competes with the e-b coupling for control of the impurity, again suggesting the possibility of a QPT.

Each of the regimes discussed above features competition between band-mediated tunneling within the manifold of impurity states and the localizing effect of the bosonic bath. Although the tunneling is dominated by a different process in the three regimes, it always drives the system towards a nondegenerate impurity ground state, whereas the e-b coupling favors a doubly-degenerate (, 2) impurity ground state. In order to provide a unified picture of the three regimes (and the regions of the parameter space that lie in between them), we will find it useful to interpret our NRG result in terms of an overall tunneling rate , which has a bare value

(50) |

Here, is assumed to be negligibly small in the local-moment regime, and to be similarly negligible in the local-charge regime. If renormalizes to large values while the e-b coupling scales to weak coupling, then one expects to recover the strong-coupling physics of the Anderson model. If, on the other hand, becomes strong while becomes weak, the system should enter a low-energy regime in which the bath governs the asymptotic low-energy, long-time impurity dynamics. Whether or not each of these scenarios is realized in practice, and whether or not there are any other possible ground states of the model, can be determined only by more detailed study. These questions are answered by the NRG results reported in the sections that follow.

## Iv Results: Symmetric model with sub-Ohmic dissipation

This section presents results for Hamiltonian (1) with and with sub-Ohmic dissipation characterized by an exponent . Figure 3 shows a schematic phase diagram on the – plane at fixed . There are two stable phases: the localized phase, in which the impurity dynamics are controlled by the coupling to the bosonic bath and the system has a pair of ground states related to one another by a particle-hole transformation; and the Kondo phase, in which there is a nondegenerate ground state. These phases are separated by a continuous QPT that takes place on the phase boundary (solid line in Fig. 3), which we parametrize as . Within the Kondo phase, the nature of the correlations evolves continuously with increasing (at fixed ) from a pure spin-Kondo effect for to a predominantly charge-Kondo effect beyond a crossover (dashed line in Fig. 3) associated with the change in sign of defined in Eq. (24).

As decreases, and the e-b coupling becomes increasingly relevant—in a renormalization-group sense [see Eq. (28)]—the phase boundary moves to the left as the localized phase grows at the expense of the Kondo phase, which disappears entirely for . As will be seen in Sec. V, the phase diagram of the Ohmic () problem has the same topology as Fig. 3, even though (as described in Sec. V) the nature of the QPT is qualitatively different than for . For , the e-b coupling is irrelevant, and the system is in the Kondo phase for all .

The remainder of this section presents the evidence for the previous statements. We first discuss the renormalization-group flows and fixed points. We then turn to the behavior in the vicinity of the phase boundary, focusing in particular on the critical response of the impurity charge to a local electric potential. Following that, we present results for the impurity spectral function, and show that the low-energy scale extracted from this spectral function supports the qualitative picture laid out in the paragraphs above and summarized in Fig. 3.

### iv.1 NRG flows and fixed points

Figure 4 plots the schematic renormalization-group flows of the couplings entering Eq. (15) and defined in Eq. (50) for a symmetric impurity () coupled to bath described by an exponent . These flows are deduced from the evolution of the many-body spectrum with increasing iteration number , i.e., with reduction in the effective band and bath cutoffs . A separatrix (dashed line) forms the boundary between the basins of attraction of a pair of stable fixed points, regions that correspond to the two phases shown in Fig. 3. Figure 4 also shows three unstable fixed points. In contrast to the situation at other points on the flow diagram, each of the fixed points exhibits a many-body spectrum that can be interpreted as the direct product of a set of bosonic excitations and a set of fermionic excitations.

The Kondo fixed point corresponds in the renormalization-group language of Fig. 4 to effective couplings and . The many-body spectrum decomposes into the direct product of (i) the excitations of a free bosonic chain described by Eq. (13) alone, and (ii) the strong-coupling excitations of the Kondo (or symmetric Anderson) model, corresponding to free electrons with a Fermi-level phase shift of . This spectrum, which exhibits SU(2) symmetry both in the spin and charge (isospin) sectors, is identical to that found throughout the Kondo phase of the particle-hole-symmetric Ising BFK HamiltonianGlossop:05 ; Glossop:07 (a model in which the bosons couple to the impurity’s spin rather than its charge).

The schematic RG flow diagram in Fig. 4 shows a localized fixed point corresponding to and . However, this is really a line of fixed points described by [Eq. (49)] with effective couplings , , and . Since , the impurity occupancy takes a fixed value or 2. (It is important to distinguish , used to characterize the fixed-point excitations, from the physical expectation value of . The latter quantity is discussed in Sec. IV.5.1.)

Each fixed point along the localized line has an excitation spectrum that decomposes into the direct product of (i) bosonic excitations identical to those at the localized fixed point of the spin-boson modelBulla:03+05 with the same bath exponent , and (ii) fermionic excitations described by a Hamiltonian

(51) |

which is just the discretized version of [Eq. (41)] with and the operator replaced by the parameter . The low-lying many-body eigenstates of appear in degenerate pairs, one member of each pair corresponding to and the other to . The fixed-point coupling increases monotonically as the bare e-b coupling decreases from infinity, and diverges on approach to the phase boundary. As illustrated in Fig. 5, this divergence can be fitted to the power-law form

(52) |

For reasons that will be explained in Sec. IV.5.1, the numerical value of coincides, to within a small error, with that of the order-parameter exponent defined in Eq. (71).

The free-orbital fixed point (, ) is unstable with respect to a bare or any deviation of from . The local-moment fixed point (), at which the impurity has a spin- degree of freedom decoupled from the band and from the bath, is reached only for bare couplings (hence, ) and .

Of greatest interest is the unstable critical fixed point that is reached for any bare couplings lying on the boundary between the Kondo and localized phases. At this fixed point, the low-lying spectrum can be constructed as the direct product of (i) the critical spectrum of the spin-boson model with the same bath exponent , and (ii) the strong-coupling spectrum of the Kondo (or symmetric Anderson) model. This spectrum, which exhibits full SU(2) symmetry in both the spin and isospin sectors, is identical to that at the critical point of the Ising-anisotropic Bose-Fermi Kondo model,critical_decomposition and is illustrated in Fig. 3(c) of Ref. Glossop:07, .

The decomposition of the critical spectrum can be understood by considering the flow of couplings entering the local-charge Hamiltonian defined in Eq. (49). The fixed-point value of the density-density coupling is in the charge-Kondo regime of the Kondo phase and diverges according to Eq. (52) in the localized phase. It is therefore reasonable to assume that in the vicinity of the phase boundary, rapidly renormalizes to strong coupling, locking the impurity site and site of the fermionic chain into one of just two states, which we can write in a pseudospin notation as and , where is the no-particle vacuum. Hopping of electrons on or off site is forbidden, so the discretized form of reduces to an effective Hamiltonian

(53) |

Here, [Eq. (34)] acts only on fermionic chain sites , and yields the Kondo/Anderson strong-coupling excitation spectrum, while

(54) |

acts on the remaining degrees of freedom in the problem in a subspace of states all carrying quantum numbers . is precisely the discretized form of the spin-boson Hamiltonian with tunneling rate and dissipation strength . These two couplings compete with one another, with three possible outcomes: (1) can scale to infinity and to zero, resulting in flow to the delocalized fixed point (the Kondo fixed point of the charge-coupled BFA model); (2) can scale to infinity and to zero, yielding flow to the localized fixed point; or (3) both couplings can renormalize to finite values , at the critical point. This picture implies that the universal critical behavior of the charge-coupled BFA model should be identical to that of the spin-boson model, the conduction-band electrons serving only to dress the impurity levels and to renormalize the impurity tunneling rate and the dissipation strength.